Question

Find an explicit form f(n) for each of the following arithmetic sequences (assume a is some real number and x is

Answer 1

Answer:

The explicit form for this sequence is $f(n)=f(1)-\frac{n-1}{10}$ for all $n\geq 1$.

Step-by-step explanation:

The explicit form of an arithmetic sequence  of numbers   is given by the formula  $f(n)=f(1)+(n-1)d$, where  $f(1)$  is the first term of the sequence,  $d$  is the difference between two consecutive terms of the sequence, and $n\geq 1$.

We know that  the first four elements for the arithmetic sequence are ${f(1)=\frac{1}{5},f(2)=\frac{1}{10},f(3)=0, f(4)=-\frac{1}{10}}$.

To find the general formula for this problem we only need to calculate $d$ in the above formula.

For n=2, we have

$f(2)=f(1)+(2-1)d=f(1)+d$

if we replace $f(1)=\frac{1}{5}$ and $f(2)=\frac{1}{10}$ and solve for $d$ we obtain  

$\frac{1}{10}=\frac{1}{5}+d$

$d=\frac{1}{10}-\frac{1}{5}=-\frac{1}{10}$

Therefore the explicit form is  $f(n)=f(1)-\frac{n-1}{10}$ for all  $n\geq 1$.